Algebraic number field

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In mathematics, an algebraic number field (or simply number field) is an extension field of the field of rational numbers such that the field extension has finite degree (and hence is an algebraic field extension). Thus is a field that contains and has finite dimension when considered as a vector space over .

The study of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory.

DefinitionEdit

PrerequisitesEdit

The notion of algebraic number field relies on the concept of a field. A field consists of a set of elements together with two operations, namely addition, and multiplication, and some distributivity assumptions. A prominent example of a field is the field of rational numbers, commonly denoted  , together with its usual operations of addition and multiplication.

Another notion needed to define algebraic number fields is vector spaces. To the extent needed here, vector spaces can be thought of as consisting of sequences (or tuples)

(x1, x2, …)

whose entries are elements of a fixed field, such as the field  . Any two such sequences can be added by adding the entries one per one. Furthermore, any sequence can be multiplied by a single element c of the fixed field. These two operations known as vector addition and scalar multiplication satisfy a number of properties that serve to define vector spaces abstractly. Vector spaces are allowed to be "infinite-dimensional", that is to say that the sequences constituting the vector spaces are of infinite length. If, however, the vector space consists of finite sequences

(x1, x2, …, xn),

the vector space is said to be of finite dimension, n.

DefinitionEdit

An algebraic number field (or simply number field) is a finite-degree field extension of the field of rational numbers. Here degree means the dimension of the field as a vector space over  .

ExamplesEdit

  • The smallest and most basic number field is the field   of rational numbers. Many properties of general number fields are modeled after the properties of  .
  • The Gaussian rationals, denoted   (read as "  adjoined  "), form the first nontrivial example of a number field. Its elements are expressions of the form
     
    where both a and b are rational numbers and i is the imaginary unit. Such expressions may be added, subtracted, and multiplied according to the usual rules of arithmetic and then simplified using the identity
     .
    Explicitly,
     
    Non-zero Gaussian rational numbers are invertible, which can be seen from the identity
     
    It follows that the Gaussian rationals form a number field which is two-dimensional as a vector space over  .
  • More generally, for any square-free integer  , the quadratic field   is a number field obtained by adjoining the square root of   to the field of rational numbers. Arithmetic operations in this field are defined in analogy with the case of Gaussian rational numbers,  .
  • Cyclotomic field
     , where  
    is a number field obtained from   by adjoining a primitive  th root of unity  . This field contains all complex nth roots of unity and its dimension over   is equal to  , where   is the Euler totient function.
  • The real numbers,  , and the complex numbers,  , are fields which have infinite dimension as  -vector spaces, hence, they are not number fields. This follows from the uncountability of   and   as sets, whereas every number field is necessarily countable.
  • The set   of ordered pairs of rational numbers, with the entrywise addition and multiplication is a two-dimensional commutative algebra over  . However, it is not a field, since it has zero divisors:
     

Algebraicity and ring of integersEdit

Generally, in abstract algebra, a field extension   is algebraic if every element   of the bigger field   is the zero of a polynomial with coefficients   in  :

 

Every field extension of finite degree is algebraic. (Proof: for   in  , simply consider   – we get a linear dependence, i.e. a polynomial that   is a root of.) In particular this applies to algebraic number fields, so any element   of an algebraic number field   can be written as a zero of a polynomial with rational coefficients. Therefore, elements of   are also referred to as algebraic numbers. Given a polynomial   such that  , it can be arranged such that the leading coefficient   is one, by dividing all coefficients by it, if necessary. A polynomial with this property is known as a monic polynomial. In general it will have rational coefficients. If, however, its coefficients are actually all integers,   is called an algebraic integer. Any (usual) integer   is an algebraic integer, as it is the zero of the linear monic polynomial:

 .

It can be shown that any algebraic integer that is also a rational number must actually be an integer, hence the name "algebraic integer". Again using abstract algebra, specifically the notion of a finitely generated module, it can be shown that the sum and the product of any two algebraic integers is still an algebraic integer. It follows that the algebraic integers in   form a ring denoted   called the ring of integers of  . It is a subring of (that is, a ring contained in)  . A field contains no zero divisors and this property is inherited by any subring, so the ring of integers of   is an integral domain. The field   is the field of fractions of the integral domain  . This way one can get back and forth between the algebraic number field   and its ring of integers  . Rings of algebraic integers have three distinctive properties: firstly,   is an integral domain that is integrally closed in its field of fractions  . Secondly,   is a Noetherian ring. Finally, every nonzero prime ideal of   is maximal or, equivalently, the Krull dimension of this ring is one. An abstract commutative ring with these three properties is called a Dedekind ring (or Dedekind domain), in honor of Richard Dedekind, who undertook a deep study of rings of algebraic integers.

Unique factorizationEdit

For general Dedekind rings, in particular rings of integers, there is a unique factorization of ideals into a product of prime ideals. For example, the ideal   in the ring   of quadratic integers factors into prime ideals as

 

However, unlike   as the ring of integers of  , the ring of integers of a proper extension of   need not admit unique factorization of numbers into a product of prime numbers or, more precisely, prime elements. This happens already for quadratic integers, for example in  , the uniqueness of the factorization fails:

 

Using the norm it can be shown that these two factorization are actually inequivalent in the sense that the factors do not just differ by a unit in  . Euclidean domains are unique factorization domains; for example  , the ring of Gaussian integers, and  , the ring of Eisenstein integers, where   is a cube root of unity (unequal to 1), have this property.[1]

ζ-functions, L-functions and class number formulaEdit

The failure of unique factorization is measured by the class number, commonly denoted h, the cardinality of the so-called ideal class group. This group is always finite. The ring of integers   possesses unique factorization if and only if it is a principal ring or, equivalently, if   has class number 1. Given a number field, the class number is often difficult to compute. The class number problem, going back to Gauss, is concerned with the existence of imaginary quadratic number fields (i.e.,  ) with prescribed class number. The class number formula relates h to other fundamental invariants of  . It involves the Dedekind zeta function ζ (s), a function in a complex variable s, defined by

 

(The product is over all prime ideals of  ,   denotes the norm of the prime ideal or, equivalently, the (finite) number of elements in the residue field  . The infinite product converges only for Re(s) > 1, in general analytic continuation and the functional equation for the zeta-function are needed to define the function for all s). The Dedekind zeta-function generalizes the Riemann zeta-function in that ζ (s) = ζ(s).

The class number formula states that ζ (s) has a simple pole at s = 1 and at this point the residue is given by

 

Here r1 and r2 classically denote the number of real embeddings and pairs of complex embeddings of  , respectively. Moreover, Reg is the regulator of  , w the number of roots of unity in   and D is the discriminant of  .

Dirichlet L-functions   are a more refined variant of  . Both types of functions encode the arithmetic behavior of   and  , respectively. For example, Dirichlet's theorem asserts that in any arithmetic progression

 

with coprime   and  , there are infinitely many prime numbers. This theorem is implied by the fact that the Dirichlet  -function is nonzero at  . Using much more advanced techniques including algebraic K-theory and Tamagawa measures, modern number theory deals with a description, if largely conjectural (see Tamagawa number conjecture), of values of more general L-functions.[2]

Bases for number fieldsEdit

Integral basisEdit

An integral basis for a number field   of degree   is a set

B = {b1, …, bn}

of n algebraic integers in   such that every element of the ring of integers   of   can be written uniquely as a Z-linear combination of elements of B; that is, for any x in   we have

x = m1b1 + ⋯ + mnbn,

where the mi are (ordinary) integers. It is then also the case that any element of   can be written uniquely as

m1b1 + ⋯ + mnbn,

where now the mi are rational numbers. The algebraic integers of   are then precisely those elements of   where the mi are all integers.

Working locally and using tools such as the Frobenius map, it is always possible to explicitly compute such a basis, and it is now standard for computer algebra systems to have built-in programs to do this.

Power basisEdit

Let   be a number field of degree  . Among all possible bases of   (seen as a  -vector space), there are particular ones known as power bases, that are bases of the form

 

for some element  . By the primitive element theorem, there exists such an  , called a primitive element. If   can be chosen in   and such that   is a basis of   as a free Z-module, then   is called a power integral basis, and the field   is called a monogenic field. An example of a number field that is not monogenic was first given by Dedekind. His example is the field obtained by adjoining a root of the polynomial[3]

 

Regular representation, trace and discriminantEdit

Recall that any field extension   has a unique  -vector space structure. Using the multiplication in  , an element   of the field   over the base field   may be represented by   matrices

 

by requiring

 

Here   is a fixed basis for  , viewed as a  -vector space. The rational numbers   are uniquely determined by   and the choice of a basis since any element of   can be uniquely represented as a linear combination of the basis elements. This way of associating a matrix to any element of the field   is called the regular representation. The square matrix   represents the effect of multiplication by   in the given basis. It follows that if the element   of   is represented by a matrix  , then the product   is represented by the matrix product  . Invariants of matrices, such as the trace, determinant, and characteristic polynomial, depend solely on the field element   and not on the basis. In particular, the trace of the matrix   is called the trace of the field element   and denoted  , and the determinant is called the norm of x and denoted  .

Now this can be generalized slightly by instead considering a field extension   and giving an  -basis for  . Then, there is an associated matrix   which has trace   and norm   defined as the trace and determinant of the matrix  .

ExampleEdit

Consider the field extension   where  . Then, we have a  -basis given by

 

since any   can be expressed as some  -linear combination

 

Then, we can take some   where   and compute  . Writing this out gives

 

We can find the matrix   by writing out the associated matrix equation giving

 

showing

 

We can then compute the trace and determinant with relative ease, giving the trace and norm.

PropertiesEdit

By definition, standard properties of traces and determinants of matrices carry over to Tr and N: Tr(x) is a linear function of x, as expressed by Tr(x + y) = Tr(x) + Tr(y), Tr(λx) = λ Tr(x), and the norm is a multiplicative homogeneous function of degree n: N(xy) = N(x) N(y), N(λx) = λn N(x). Here λ is a rational number, and x, y are any two elements of  .

The trace form derived is a bilinear form defined by means of the trace, as

  by  

 . The integral trace form, an integer-valued symmetric matrix is defined as  , where b1, …, bn is an integral basis for  . The discriminant of   is defined as det(t). It is an integer, and is an invariant property of the field  , not depending on the choice of integral basis.

The matrix associated to an element x of   can also be used to give other, equivalent descriptions of algebraic integers. An element x of   is an algebraic integer if and only if the characteristic polynomial pA of the matrix A associated to x is a monic polynomial with integer coefficients. Suppose that the matrix A that represents an element x has integer entries in some basis e. By the Cayley–Hamilton theorem, pA(A) = 0, and it follows that pA(x) = 0, so that x is an algebraic integer. Conversely, if x is an element of   which is a root of a monic polynomial with integer coefficients then the same property holds for the corresponding matrix A. In this case it can be proven that A is an integer matrix in a suitable basis of  . The property of being an algebraic integer is defined in a way that is independent of a choice of a basis in  .

Example with integral basisEdit

Consider  , where x satisfies x3 − 11x2 + x + 1 = 0. Then an integral basis is [1, x, 1/2(x2 + 1)], and the corresponding integral trace form is

 

The "3" in the upper left hand corner of this matrix is the trace of the matrix of the map defined by the first basis element (1) in the regular representation of   on  . This basis element induces the identity map on the 3-dimensional vector space,  . The trace of the matrix of the identity map on a 3-dimensional vector space is 3.

The determinant of this is 1304 = 23·163, the field discriminant; in comparison the root discriminant, or discriminant of the polynomial, is 5216 = 25·163.

PlacesEdit

Mathematicians of the nineteenth century assumed that algebraic numbers were a type of complex number.[4][5] This situation changed with the discovery of p-adic numbers by Hensel in 1897; and now it is standard to consider all of the various possible embeddings of a number field   into its various topological completions   at once.

A place of a number field   is an equivalence class of absolute values on  [6]pg 9. Essentially, an absolute value is a notion to measure the size of elements   of  . Two such absolute values are considered equivalent if they give rise to the same notion of smallness (or proximity). The equivalence relation between absolute values   is given by some   such that

 
meaning we take the value of the norm   to the  -th power.

In general, the types of places fall into three regimes. Firstly (and mostly irrelevant), the trivial absolute value | |0, which takes the value   on all non-zero  . The second and third classes are Archimedean places and non-Archimedean (or ultrametric) places. The completion of   with respect to a place   is given in both cases by taking Cauchy sequences in  and dividing out null sequences, that is, sequences   such that

 
tends to zero when   tends to infinity. This can be shown to be a field again, the so-called completion of   at the given place  , denoted  .

For  , the following non-trivial norms occur (Ostrowski's theorem): the (usual) absolute value, sometimes denoted   which gives rise to the complete topological field of the real numbers  . On the other hand, for any prime number  , the p-adic absolute values is defined by

|q|p = pn, where q = pn a/b and a and b are integers not divisible by p.

are used to construct the  -adic numbers  . In contrast to the usual absolute value, the p-adic norm gets smaller when q is multiplied by p, leading to quite different behavior of   vis-à-vis  .

Note the general situation typically considered is taking a number field   and considering a prime ideal   for its associated ring of algebraic numbers  . Then, there will be a unique place   called a non-archimedian place. In addition, for every embedding   there will be a place called an archimedian place, denoted  . This statement is a theorem also called Ostrowski's theorem.

ExamplesEdit

The field   for   where   is a fixed 6th root of unity, provides a rich example for constructing explicit real and complex archimedian embeddings, and non-archimedian embeddings as well[6]pg 15-16.

Archimedean placesEdit

Here we use the standard notation   and   for the number of real and complex embeddings used, respectively (see below).

Calculating the archimedean places of a number field   is done as follows: let   be a primitive element of  , with minimal polynomial   (over  ). Over  ,   will generally no longer be irreducible, but its irreducible (real) factors are either of degree one or two. Since there are no repeated roots, there are no repeated factors. The roots   of factors of degree one are necessarily real, and replacing   by   gives an embedding of   into  ; the number of such embeddings is equal to the number of real roots of  . Restricting the standard absolute value on   to   gives an archimedean absolute value on  ; such an absolute value is also referred to as a real place of  . On the other hand, the roots of factors of degree two are pairs of conjugate complex numbers, which allows for two conjugate embeddings into  . Either one of this pair of embeddings can be used to define an absolute value on  , which is the same for both embeddings since they are conjugate. This absolute value is called a complex place of  .[7][8]

If all roots of   above are real (respectively, complex) or, equivalently, any possible embedding   is actually forced to be inside   (resp.  ),   is called totally real (resp. totally complex).[9][10]

Non-Archimedean or ultrametric placesEdit

To find the non-Archimedean places, let again   and   be as above. In  ,   splits in factors of various degrees, none of which are repeated, and the degrees of which add up to  , the degree of  . For each of these  -adically irreducible factors  , we may suppose that   satisfies   and obtain an embedding of   into an algebraic extension of finite degree over  . Such a local field behaves in many ways like a number field, and the  -adic numbers may similarly play the role of the rationals; in particular, we can define the norm and trace in exactly the same way, now giving functions mapping to  . By using this  -adic norm map   for the place  , we may define an absolute value corresponding to a given  -adically irreducible factor   of degree   by

 
Such an absolute value is called an ultrametric, non-Archimedean or  -adic place of  .

For any ultrametric place v we have that |x|v ≤ 1 for any x in  , since the minimal polynomial for x has integer factors, and hence its p-adic factorization has factors in Zp. Consequently, the norm term (constant term) for each factor is a p-adic integer, and one of these is the integer used for defining the absolute value for v.

Prime ideals in OKEdit

For an ultrametric place v, the subset of   defined by |x|v < 1 is an ideal   of  . This relies on the ultrametricity of v: given x and y in  , then

|x + y|v ≤ max (|x|v, |y|v) < 1.

Actually,   is even a prime ideal.

Conversely, given a prime ideal   of  , a discrete valuation can be defined by setting   where n is the biggest integer such that  , the n-fold power of the ideal. This valuation can be turned into an ultrametric place. Under this correspondence, (equivalence classes) of ultrametric places of   correspond to prime ideals of  . For  , this gives back Ostrowski's theorem: any prime ideal in Z (which is necessarily by a single prime number) corresponds to a non-Archimedean place and vice versa. However, for more general number fields, the situation becomes more involved, as will be explained below.

Yet another, equivalent way of describing ultrametric places is by means of localizations of  . Given an ultrametric place   on a number field  , the corresponding localization is the subring   of   of all elements   such that | x |v ≤ 1. By the ultrametric property   is a ring. Moreover, it contains  . For every element x of  , at least one of x or x−1 is contained in  . Actually, since K×/T× can be shown to be isomorphic to the integers,   is a discrete valuation ring, in particular a local ring. Actually,   is just the localization of   at the prime ideal  , so  . Conversely,   is the maximal ideal of  .

Altogether, there is a three-way equivalence between ultrametric absolute values, prime ideals, and localizations on a number field.

Lying over theorem and placesEdit

One of the basic theorems in algebraic number theory are the going up and going down theorems, which describe the behavior of some prime ideal   when it is extended as an ideal in   for some field extension  . We say that an ideal   lies over   if  . Then, one incarnation of the theorem states a prime ideal in   lies over  , hence there is always a surjective map

 
induced from the inclusion  . Since there exists a correspondence between places and prime ideals, this means we can find places dividing a place which is induced from a field extension. That is, if   is a place of  , then there are places   of   which divide  , in the sense that their induced prime ideals divide the induced prime ideal of   in  . In fact, this observation is useful[6]pg 13 while looking at the base change of an algebraic field extension of   to one of its completions  . If we write
 
and write   for the induced element of  , we get a decomposition of  . Explicitly, this decomposition is
 
furthermore, the induced polynomial   decomposes as
 
because of Hensel's lemma[11]pg 129-131, hence
 
Moreover, there are embeddings
 
where   is a root of   giving  , hence we could write
 
as subsets of   (which is the completion of the algebraic closure of  ).

RamificationEdit

 
Schematic depiction of ramification: the fibers of almost all points in Y below consist of three points, except for two points in Y marked with dots, where the fibers consist of one and two points (marked in black), respectively. The map f is said to be ramified in these points of Y.

Ramification, generally speaking, describes a geometric phenomenon that can occur with finite-to-one maps (that is, maps   such that the preimages of all points y in Y consist only of finitely many points): the cardinality of the fibers f−1(y) will generally have the same number of points, but it occurs that, in special points y, this number drops. For example, the map

 

has n points in each fiber over t, namely the n (complex) roots of t, except in t = 0, where the fiber consists of only one element, z = 0. One says that the map is "ramified" in zero. This is an example of a branched covering of Riemann surfaces. This intuition also serves to define ramification in algebraic number theory. Given a (necessarily finite) extension of number fields  , a prime ideal p of   generates the ideal pOK of  . This ideal may or may not be a prime ideal, but, according to the Lasker–Noether theorem (see above), always is given by

pO  = q1e1 q2e2qmem

with uniquely determined prime ideals qi of   and numbers (called ramification indices) ei. Whenever one ramification index is bigger than one, the prime p is said to ramify in  .

The connection between this definition and the geometric situation is delivered by the map of spectra of rings  . In fact, unramified morphisms of schemes in algebraic geometry are a direct generalization of unramified extensions of number fields.

Ramification is a purely local property, i.e., depends only on the completions around the primes p and qi. The inertia group measures the difference between the local Galois groups at some place and the Galois groups of the involved finite residue fields.

An exampleEdit

The following example illustrates the notions introduced above. In order to compute the ramification index of  , where

f(x) = x3x − 1 = 0,

at 23, it suffices to consider the field extension  . Up to 529 = 232 (i.e., modulo 529) f can be factored as

f(x) = (x + 181)(x2 − 181x − 38) = gh.

Substituting x = y + 10 in the first factor g modulo 529 yields y + 191, so the valuation | y |g for y given by g is | −191 |23 = 1. On the other hand, the same substitution in h yields y2 − 161y − 161 modulo 529. Since 161 = 7 × 23,

 

Since possible values for the absolute value of the place defined by the factor h are not confined to integer powers of 23, but instead are integer powers of the square root of 23, the ramification index of the field extension at 23 is two.

The valuations of any element of   can be computed in this way using resultants. If, for example y = x2x − 1, using the resultant to eliminate x between this relationship and f = x3x − 1 = 0 gives y3 − 5y2 + 4y − 1 = 0. If instead we eliminate with respect to the factors g and h of f, we obtain the corresponding factors for the polynomial for y, and then the 23-adic valuation applied to the constant (norm) term allows us to compute the valuations of y for g and h (which are both 1 in this instance.)

Dedekind discriminant theoremEdit

Much of the significance of the discriminant lies in the fact that ramified ultrametric places are all places obtained from factorizations in   where p divides the discriminant. This is even true of the polynomial discriminant; however the converse is also true, that if a prime p divides the discriminant, then there is a p-place which ramifies. For this converse the field discriminant is needed. This is the Dedekind discriminant theorem. In the example above, the discriminant of the number field   with x3 − x − 1 = 0 is −23, and as we have seen the 23-adic place ramifies. The Dedekind discriminant tells us it is the only ultrametric place which does. The other ramified place comes from the absolute value on the complex embedding of  .

Galois groups and Galois cohomologyEdit

Generally in abstract algebra, field extensions K / L can be studied by examining the Galois group Gal(K / L), consisting of field automorphisms of   leaving   elementwise fixed. As an example, the Galois group   of the cyclotomic field extension of degree n (see above) is given by (Z/nZ)×, the group of invertible elements in Z/nZ. This is the first stepstone into Iwasawa theory.

In order to include all possible extensions having certain properties, the Galois group concept is commonly applied to the (infinite) field extension K / K of the algebraic closure, leading to the absolute Galois group G := Gal(K / K) or just Gal(K), and to the extension  . The fundamental theorem of Galois theory links fields in between   and its algebraic closure and closed subgroups of Gal(K). For example, the abelianization (the biggest abelian quotient) Gab of G corresponds to a field referred to as the maximal abelian extension Kab (called so since any further extension is not abelian, i.e., does not have an abelian Galois group). By the Kronecker–Weber theorem, the maximal abelian extension of   is the extension generated by all roots of unity. For more general number fields, class field theory, specifically the Artin reciprocity law gives an answer by describing Gab in terms of the idele class group. Also notable is the Hilbert class field, the maximal abelian unramified field extension of  . It can be shown to be finite over  , its Galois group over   is isomorphic to the class group of  , in particular its degree equals the class number h of   (see above).

In certain situations, the Galois group acts on other mathematical objects, for example a group. Such a group is then also referred to as a Galois module. This enables the use of group cohomology for the Galois group Gal(K), also known as Galois cohomology, which in the first place measures the failure of exactness of taking Gal(K)-invariants, but offers deeper insights (and questions) as well. For example, the Galois group G of a field extension L / K acts on L×, the nonzero elements of L. This Galois module plays a significant role in many arithmetic dualities, such as Poitou-Tate duality. The Brauer group of  , originally conceived to classify division algebras over  , can be recast as a cohomology group, namely H2(Gal (K, K×).

Local-global principleEdit

Generally speaking, the term "local to global" refers to the idea that a global problem is first done at a local level, which tends to simplify the questions. Then, of course, the information gained in the local analysis has to be put together to get back to some global statement. For example, the notion of sheaves reifies that idea in topology and geometry.

Local and global fieldsEdit

Number fields share a great deal of similarity with another class of fields much used in algebraic geometry known as function fields of algebraic curves over finite fields. An example is Kp(T). They are similar in many respects, for example in that number rings are one-dimensional regular rings, as are the coordinate rings (the quotient fields of which is the function field in question) of curves. Therefore, both types of field are called global fields. In accordance with the philosophy laid out above, they can be studied at a local level first, that is to say, by looking at the corresponding local fields. For number fields  , the local fields are the completions of   at all places, including the archimedean ones (see local analysis). For function fields, the local fields are completions of the local rings at all points of the curve for function fields.

Many results valid for function fields also hold, at least if reformulated properly, for number fields. However, the study of number fields often poses difficulties and phenomena not encountered in function fields. For example, in function fields, there is no dichotomy into non-archimedean and archimedean places. Nonetheless, function fields often serves as a source of intuition what should be expected in the number field case.

Hasse principleEdit

A prototypical question, posed at a global level, is whether some polynomial equation has a solution in  . If this is the case, this solution is also a solution in all completions. The local-global principle or Hasse principle asserts that for quadratic equations, the converse holds, as well. Thereby, checking whether such an equation has a solution can be done on all the completions of  , which is often easier, since analytic methods (classical analytic tools such as intermediate value theorem at the archimedean places and p-adic analysis at the nonarchimedean places) can be used. This implication does not hold, however, for more general types of equations. However, the idea of passing from local data to global ones proves fruitful in class field theory, for example, where local class field theory is used to obtain global insights mentioned above. This is also related to the fact that the Galois groups of the completions Kv can be explicitly determined, whereas the Galois groups of global fields, even of   are far less understood.

Adeles and idelesEdit

In order to assemble local data pertaining to all local fields attached to  , the adele ring is set up. A multiplicative variant is referred to as ideles.

See alsoEdit

GeneralizationsEdit

Algebraic number theoryEdit

Class field theoryEdit

NotesEdit

  1. ^ Ireland, Kenneth; Rosen, Michael (1998), A Classical Introduction to Modern Number Theory, Berlin, New York: Springer-Verlag, ISBN 978-0-387-97329-6, Ch. 1.4
  2. ^ Bloch, Spencer; Kato, Kazuya (1990), "L-functions and Tamagawa numbers of motives", The Grothendieck Festschrift, Vol. I, Progr. Math., 86, Boston, MA: Birkhäuser Boston, pp. 333–400, MR 1086888
  3. ^ Narkiewicz 2004, §2.2.6
  4. ^ Kleiner, Israel (1999), "Field theory: from equations to axiomatization. I", The American Mathematical Monthly, 106 (7): 677–684, doi:10.2307/2589500, MR 1720431, To Dedekind, then, fields were subsets of the complex numbers.
  5. ^ Mac Lane, Saunders (1981), "Mathematical models: a sketch for the philosophy of mathematics", The American Mathematical Monthly, 88 (7): 462–472, doi:10.2307/2321751, MR 0628015, Empiricism sprang from the 19th-century view of mathematics as almost coterminal with theoretical physics.
  6. ^ a b c Gras, Georges (2003). Class field theory : from theory to practice. Berlin. ISBN 978-3-662-11323-3. OCLC 883382066.
  7. ^ Cohn, Chapter 11 §C p. 108
  8. ^ Conrad
  9. ^ Cohn, Chapter 11 §C p. 108
  10. ^ Conrad
  11. ^ Neukirch, Jürgen (1999). Algebraic Number Theory. Berlin, Heidelberg: Springer Berlin Heidelberg. ISBN 978-3-662-03983-0. OCLC 851391469.

ReferencesEdit